I am reading "Topics in Algebra 2nd Edition" by I. N. Herstein.
The following problem is Problem 19 on p.71 in Herstein's book.
Let $G$ be a group and $T$ an automorphism of $G$. If $N$ is a normal subgroup of $G$ such that $(N)T\subset N$, show how you could use $T$ to define an automorphism of $G/N$.
The following answer is the best answer for this problem.
https://math.stackexchange.com/a/2763736/384082
But I think that this answer assumes $(N)T=N$.
Can we change the assumption $(N)T\subset N$ to $(N)T=N$?
If we cannot change the assumption $(N)T\subset N$ to $(N)T=N$, please tell me an answer for this problem.
I am very poor at English, so maybe I misunderstand what this problem says.
My attempt:
$G/(N)T\approx G/N$.
$G/(N)T^{-1}\approx G/N$.
If $\# N=+\infty$, I guess there is an example such that there is an element $x\in G$ such that $x\notin N$ and $(x)T\in N$.
I guess $\overline{T}:G/N\ni\overline{g}\mapsto\overline{(g)T}\in G/N$ is not injective in general.